When one performs a assist you only get the takepoints for the zone, no pph which you only get when you do a takeover of the zone. Everyone who are standing in a zone while a takeover is done gets a assist.

The one of you who has the shortest takeover time will be the one to get the takeover if you enters the zone simultaneously. You can check out your takeover time by clicking on your ranks on the info page: https://turfgame.com/info

In the time b.a. (before assist) the player with the shortest takeover time claimed both the takeover points and the pph for owning the zone, if another player was standing in the zone with longer takeover time he would had to leave empty handed.

The player who gets the takeover will raise his arms fully and the player getting the assist will raise them half way up.

It sound like you have a shorter takeover time then your son so one solution would be for you to enter the zone a few seconds after your son.

Note also that the assister gets a take on the zone in all aspects except owning it afterwards. Same takepoints as already mentioned, +1 on your total take count, it is included in the session graph at warded, and zone is marked as visited (if you care for unique zones).